Kitap 10
4
But the greatest common measure of A, B is D; therefore F measures D.
4
But it measures C also; therefore F measures C, D; therefore F will also measure the greatest common measure of C, D. [X. 3, Por.]
4
But that is E; therefore F will measure E, the greater the less: which is impossible.
4
Therefore no magnitude greater than the magnitude E will measure A, B, C; therefore E is the greatest common measure of A, B, C if D do not measure C, and, if it measure it, D is itself the greatest common measure.
4
Therefore the greatest common measure of the three given commensurable magnitudes has been found.
4
Porism. From this it is manifest that, if a magnitude measure three magnitudes, it will also measure their greatest common measure.
4
Similarly too, with more magnitudes, the greatest common measure can be found, and the porism can be extended. Q. E. D.
PROPOSITION 5.
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Commensurable magnitudes have to one another the ratio which a number has to a number.
5
Let A, B be commensurable magnitudes; I say that A has to B the ratio which a number has to a number.
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For, since A, B are commensurable, some magnitude will measure them.
5
Let it measure them, and let it be C.
5
And, as many times as C measures A, so many units let there be in D; and, as many times as C measures B, so many units let there be in E.
5
Since then C measures A according to the units in D, while the unit also measures D according to the units in it, therefore the unit measures the number D the same number of times as the magnitude C measures A; therefore, as C is to A, so is the unit to D; [VII. Def. 20] therefore, inversely, as A is to C, so is D to the unit. [cf. V. 7, Por.]
5
Again, since C measures B according to the units in E, while the unit also measures E according to the units in it, therefore the unit measures E the same number of times as C measures B; therefore, as C is to B, so is the unit to E.
5
But it was also proved that, as A is to C, so is D to the unit; therefore, ex aequali, as A is to B, so is the number D to E. [V. 22]
